Federal Reserve Bank of Minneapolis. Research Department Staff Report. November 2009. Bailouts, Time Inconsistency and Optimal Regulation". V. V. Chari.
Federal Reserve Bank of Minneapolis Research Department Sta¤ Report November 2009
Bailouts, Time Inconsistency and Optimal Regulation V. V. Chari University of Minnesota and Federal Reserve Bank of Minneapolis
Patrick J. Kehoe Federal Reserve Bank of Minneapolis, University of Minnesota, and Princeton University
ABSTRACT We make three points. First, ex ante e¢ cient contracts often require ex post ine¢ ciency. Second, the time inconsistency problem for the government is more severe than for private agents because …re sale e¤ects give governments stronger incentives to renegotiate contracts than private agents. Third, given that the government cannot commit itself to not bailing out …rms ex post, ex ante regulation of …rms is desirable.
Chari, and Kehoe thank the National Science Foundation for …nancial support. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Minneapolis or the Federal Reserve System.
Recent experience has shown that governments can, will, and perhaps should intervene during …nancial crises. Such interventions typically occur because governments seek to minimize the spillover e¤ects of bankruptcy and liquidation upon the broader economy. Such interventions during …nancial crises alter the incentives for …rms and …nancial intermediaries ex ante. In this paper we ask how optimal regulation should be designed to maximize ex ante welfare taking into account the temptation for the government to intervene ex post. The theme that we explore in this paper is that, by altering private contracts, the prospect of bailouts reduces ex ante welfare. We view the prescription that governments should refrain from bailing out potentially bankrupt …rms as unrealistic in practice. Benevolent governments simply do not have the power to commit themselves to such a prescription. A pragmatic approach to policy dictates that we take as given the incentives of governments to undertake bailouts and design ex ante regulation to minimize the ex ante costs of these ex post bailouts. In thinking about bailouts by governments, a central question is why would the government …nd it optimal to bail out …rms ex post. We argue that confronted with an ex post situation in which many …rms are about to undergo costly bankruptcies, a benevolent government has a strong incentive to bail out …rms. These ex post bailouts, however, may distort the ex ante incentives of managers and …rms and reduce ex ante welfare. In such a situation, a government with commitment would commit itself not to undertake bailouts. If the government lacks such commitment, it will bail out …rms ex post and the expectation of such bailouts will reduce ex ante welfare. In this sense, the government has a time inconsistency problem in bailout policy. We show that this time inconsistency problem creates a role for ex ante regulation. Such regulation can reduce the temptation of governments to bail out
…rms ex post and thereby raise ex ante welfare. In analyzing the incentives of benevolent governments to intervene and prevent costly bankruptcies ex post, the obvious question arises, why would …rms ex ante enter into contracts which impose ex post costs? More generally, why would …rms design contracts that feature ex post ine¢ cient outcomes? Here we develop a model in which the optimal contract between a …rm and a manager speci…es bankruptcy when outcomes are bad in order to provide proper incentives to managers to engage in e¤ort. Bankruptcy is costly in two ways: it reduces the output of the …rm and it imposes nonpecuniary costs on the manager. We think of these nonpecuniary costs as arising both from stigma-like e¤ects on the manager’s career as well as loss of private bene…ts from operating the …rm. In the model the optimal contract is ex post ine¢ cient in the sense that, once the manager has exerted e¤ort, bankruptcy imposes costs on the owners of the …rm and the manager. While these ex post ine¢ ciencies create a time inconsistency problem for the government by giving it an incentive to bailout …rms ex post, they also create a time inconsistency problem for private agents by giving them an incentive to avoid costly bankruptcy by renegotiating their contracts ex post. Analyzing these incentives requires modeling the bene…ts and costs of both renegotiation and bailouts. The bene…ts are the reduction in costly bankruptcies. We assume that the costs arise from changes in the beliefs of private agents about future outcomes. In particular, if a …rm ever agrees to renegotiate, private agents will believe that …rm will always renegotiate in the future. Expectations of such renegotiations constrain future contracts and thereby reduce future welfare. Likewise, if a government ever bails out …rms, private agents believe that the government will always bailout …rms in the future. Expectations of such bailouts constrain future contracts and reduce future welfare. 2
In an environment without commitment, private agents and governments balance these bene…ts and costs in designing their interventions. For the private agents, this balance implies that ex ante optimal contracts must satisfy a private sustainability constraint. For the government, this balance implies that ex ante optimal contracts must, in equilibrium, satisfy a sustainability to bailouts constraint. The parallel way we have modeled bene…ts and costs for governments and private agents leads us to ask, Given that a contract has already been designed to be privately sustainable, why would it not be sustainable to bailouts? When deciding whether to renegotiate a given contract, the private agents involved in that contract consider the bene…ts from eliminating bankruptcy of their …rm at given prices. When the government decides to bail out …rms, it takes into account the private bene…ts per …rm in the same way that private agents do, but, in addition, it also takes into account the bene…ts to other …rms from its intervention. These bene…ts arise because by bailing out …rms the government can reduce the aggregate amount of assets sold in the market place and thereby raise the prices of these assets. The idea is that bankruptcy is socially costly because it forces …rms to sell their assets and these …re sales reduce the value of assets in otherwise healthy …rms. Bailouts help reduce …re sales and the resulting negative price e¤ects that give rise to the social cost. Since governments take into account …re sale e¤ects and private agents do not, the sustainability to bailouts constraint is tighter than the private sustainability constraint. Thus, a contract that is privately sustainable is not necessarily sustainable to bailouts. In this sense, the time inconsistency problem is for the government is more severe than it is for private agents. The greater severity of the time inconsistency problem for the government implies that the equilibrium in an economy with bailouts has lower welfare than in an economy without 3
bailouts. It also implies that ex-ante regulation can be desirable. Such regulation must be designed so that ex-post the government does not have an incentive to engage in bailouts. The incentive to bail out …rms is large when the aggregate amount of assets in bankrupt …rms is large. We show that the optimal ex-ante regulation is to impose a cap on quantity of assets used by each manager and a cap on the probability of bankruptcy. This cap on assets limits the size of individual …rms and thus can be interpreted as a regulation that prevents …rms from becoming too big. We refer to this regulation as a too-big-to-fail-cap. The cap on the probability of bankruptcy can be implemented by a cap on the debt to value ratio of the …rm. The reason is that this ratio is an increasing function of the probability of bankruptcy so that a cap on the probability of bankruptcy is equivalent to a cap on the debt to value ratio.
1. A simple economy We begin with a simple static version of our benchmark economy. We use this version to show that, in order to provide incentives, optimal contracts often require ex post ine¢ ciency, in the sense that ex post all agents can bene…t by altering the terms of the contract. This feature of the model makes optimal contracts time inconsistent, in the sense that optimal contracts without commitment di¤er from those with commitment, and, in particular, give lower welfare. Consider a model in which decisions are made at two stages: a …rst stage, called the beginning of the period, and a second stage called the end of the period. There are two types of agents, called lenders and managers both of whom are risk neutral and consume at the end of the period. There is a measure 1 of managers and a measure 1 of lenders.
4
The economy has two production technologies. The storage technology is available to all agents, which transforms one unit of endowments at the …rst stage into one unit of consumption goods at the second stage. The corporate technology speci…es projects that require two inputs at the …rst stage: e¤ort a of managers and an investment of 1 of goods. This technology transforms these inputs into capital goods. The capital goods then can be used to make stage two consumption goods. E¤ort a of managers is unobserved by lenders. If the corporate technology is used the amount of capital goods produced in the second stage stochastically depends on the e¤ort level a of the manager as well as an idiosyncratic exogenous shock representing the manager’s current draw of ability. In particular, given e¤ort level a and a draw of " with probability pH (a) the high state is realized and AH (1 + ") units of capital goods are produced and with probability pL (a) = 1
pH (a) the low state is realized
and AL (1 + ") units of capital goods are produced where AL < AH : We assume that higher e¤ort levels increase the probability of the high state. Speci…cally, we assume that pH (a) is an increasing, strictly concave function of a: Notice that since pH (a) is increasing this technology satis…es that monotone likelihood ratio property and since pH (a) is strictly concave it satisi…es the convexity of distribution function property1 . These assumptions guarantee that the …rst order approach is valid. (See Rogerson 19?? for details.) We assume that the mean of " is zero. We think of the project as being undertaken by a …rm. We think of managers as 1
Recall that the monotone likelihood ratio property is that if a > a ^ pH (a) 1 > pH (^ a) 1
pH (a) pH (^ a)
while the convexity of distribution property is that the cdf induced by pH (a); namely FL (a) = 1 a strictly positive second derivative.
5
pH (a); has
performing two tasks. The …rst task is to exert e¤ort a and transform consumption goods from stage 1 into capital goods at stage 2. The second task is to transform capital goods stage 2 into …nal consumption goods. After the manager has completed the …rst task and a certain amount of capital has been produced the …rm can choose to continue the project under the current manager or it can declare bankruptcy. If it continues then the project produces one unit of output for every unit of capital, so that the …rm’s output is
(1)
Yci (") = Ai (1 + ") for i 2 fH; Lg
where c denotes continue: If the …rm declares bankruptcy, the manager is removed, the …rm incurs a direct output loss and the manager su¤ers a nonpecuniary cost. The direct output loss occurs because following bankruptcy the capital Ai (1+") is used in an inferior technology, referred to as the traditional technology, that yields R
1 consumption goods for every unit
of capital invested so that the value of the output of the …rm in the bankruptcy state is
(2)
Ybi (") = RAi (1 + ")
where b denotes bankruptcy. In the event of bankruptcy the manager su¤ers a nonpecuniary loss
B: This nonpecuniary cost is supposed to represent extra costs to the manager, such
as a loss in reputation or a loss in nonpecuniary bene…ts from being employed as a manager that are incurred from a liquidation. Lenders are endowed with e units of a consumption good in the …rst stage but cannot
6
operate the corporate technology. Managers have no endowments of goods but can operate the corporate technology. Lenders choose whether to lend to …rms that operate the corporate technology or to store their endowments. We assume that e > 1. Since the economy has an equal measure of managers and lenders and since the corporate technology uses 1 unit of the endowment per manager the storage technology is always active and the rate of return to lending to the corporate technology is 1: Let ci (") denote the consumption of the managers in state i given the realization " and di (") the return to the investor in a project when the state is i and the idiosyncratic shock is given by ": Let Bi denote the set of idiosyncratic shocks " such that the …rms declares bankruptcy in state i 2 fH; Lg and Ci denote the complementary sent in which the project is continued. We assume that …rms, referred to as …nancial intermediaries, operate a continuum of projects. Given the symmetry of the expected returns across projects, …nancial intermediaries will choose the same e¤ort level for all managers. The pro…ts generated by a …nancial intermediary which …nds it optimal to operate the corporate technology at a positive level are
(3)
X i
pi (a)
Z
Ci
Yci (")dH(") +
Z
Ybi (")dH(")
Bi
Z
[ci (") + di (")]dH(")
…nancial intermediaries compete in o¤ering contracts to managers and lenders. These contracts must attract investment by lenders so that they must o¤er a return to lenders of at least one. Thus, a contract must meet the following participation constraint for lenders 7
(4)
X
pi (a)
i
Z
di (")dH(")
1
The contracts must also attract managers. Let U denote the value of the best alternative contract o¤ered to a managers. Thus, a contract must meet a participation constraint for managers
(5)
X
pi (a)
i
Z
ci (")dH(")
B
Z
dH(")
a
U:
Bi
Since the e¤ort choice a of managers is unobservable a contract must satisfy an incentive constraint given by
(6)
a 2 arg max a
X
pi (a)
i
Z
ci (")dH(")
B
Z
dH(")
a:
Bi
Finally, the consumption of managers must satisfy a nonnegativity constraint
(7)
ci (")
0
A. With commitment Suppose now that …nancial intermediaries and managers can commit to contracts. Under this assumption the …nancial intermediaries’contracting problem is to choose a recommended action a; compensation schemes ci ( ), di ( ) and bankruptcy and continuation sets Bi and Ci to maximize pro…ts (3) subject to (4), (5), (6), and (7).
8
Clearly the consumption level of a lender that lends 1 to …nancial intermediaries and invests the rest in the storage technology is given by
(8)
I
c =
X
pi (a)
i
Z
di (")dH(") + e
1
The resource constraint is
(9)
X
pi (a)
Z
ci (")dH(") + cI
pi (a)
Z
Z
i
X i
Ci
Yci (")dH(") +
Ybi (")dH(") + e
1
Bi
An allocation is a collection a; ci ( ), di ( ), cI , Ci ; Bi . A competitive equilibrium is an allocation together with a minimum utility level U such that i) the allocations a; ci ( ), di ( ), and sets Ci ; Bi solve the contracting problem. ii) the minimum utility level U is such that …rm pro…ts are zero. iii) the consumption of lenders satis…es (8). iv) the resource constraint (9) holds. Note here that U plays the role of a price and that by Walras’ Law the resource constraint is implied by zero pro…ts of …nancial intermediaries and the consumption of lenders (8). Throughout we will restrict attention to environments in which the competitive equilibrium has an active corporate technology. A su¢ cient condition for such an equilibrium to exist is that AH and p0 (0) are su¢ ciently large. We turn the e¢ ciency of a competitive equilibrium. Given a utility level of lenders cI ; 9
an allocation is e¢ cient if it satis…es the following planning problem, namely to maximize the welfare of managers subject to (6), (7), (8), and
(10) cI
cI :
Proposition 1. The competitive equilibrium is e¢ cient. Proof : Since pro…ts are zero in a competitive equilibrium, we can use duality to rewrite the contracting problem as one of maximizing the utility of managers subject to the constraint the …rm pro…ts be nonnegative. Substituting for the consumption of lenders from (8) into …nancial intermediaries’pro…ts (3) yields the resource constraint. Clearly, the rewritten contracting problem coincides with the planning problem. Q:E:D: Consider the following assumption. Let aO be the solution to the version of the problem with publicly observed e¤ort, namely the value of a that solves
(11) p0H (a)AH
AL = 1:
Assume that
(12) pH (aO ) < 1
Proposition 2. If AL < 1 and (12) holds, then the competitive equilibrium with privately observed e¤ort information has strictly lower e¤ort level a and welfare than the competitive equilibrium with publicly observed e¤ort. Proof. In the competitive equilibrium with publicly observed e¤ort it is straightforward 10
to show that the optimal e¤ort level solves (11) and the liquidation sets BH and BL are empty. The …rst order condition for e¤ort in the private information economy is X
p0i (a)
i
Z
ci (")dG(")
B
Z
dG(") = 1
Bi
A moment’s re‡ection makes clear that the only way to support the allocations with publicly observed e¤ort in the economy with privately observed e¤ort is to pay the manager an expected compensation of
(13)
Z
cH (")dH(") = AH
AL
in the high state and zero in the low state. But, since AL < 1 if …nancial intermediaries pay managers this much and pay the lenders 1 unit then pro…ts are negative. To establish this result substitute (1), (2), (4) with equality and (13) into the expression for …rm’s pro…ts (3) and using the assumption that the expected value of " is zero, to obtain
pH (a) [AH
(AH
AL )] + pL (a)AL
1 = AL
1
which is negative since AL < 1: Q:E:D: From here onwards the term competitive equilibrium refers to competitive equilibrium with privately observed e¤ort. We now show that the contracting problem reduces to a simpler one under the condition that AL < 1: We will show that in any competitive equilibrium the optimal contracting
11
problem can be reduced to the following: Choose cH ; a; and " to solve
(14) max pH (a)cH
pL (a)BH(" )
a
subject to
(15) a 2 arg max pH (a)cH a
(16) pH (a)cH + 1
pL (a)BH(" )
pH (a)AH + pL (a)AL
a: Z
"
(1 + ")dH(") + R
"
Z
"
(1 + ")dH(")
"
We refer to x = (cH ; a; " ) as the contract. To establish this result we …rst note that if AL < 1 the incentive constraint is always binding. Hence an optimal contract must reward the manager only in the high state and set the consumption of managers in the low state to be zero for all "; that is, cL (") = 0: The intuition for this result is that as long as consumption is positive in the low state, manager’s incentives can be improved by shifting consumption from the low state to the high state. Since the manager cares only about expected consumption the optimum can be achieved by setting consumption in the high state to be a constant so that cH (") = cH . Second, note the only role of bankruptcy is to improve incentives so that it is never optimal to declare bankruptcy in the high state. In the low state, the optimal bankruptcy rule has a cuto¤ form: declare bankruptcy for " follows because the output loss from bankruptcy, (1
" and continue otherwise. This result R)AL (1 + "); is smaller the lower is
" and the manager only cares about the probability of bankruptcy in the low state. More formally, if the optimal contract had bankruptcy for a high realization " and continuation 12
for a low realization of "; then the output loss could be reduced by rearranging the set of realizations for which there is bankruptcy while maintaining the manager’s incentives. Third, in any competitive equilibrium pro…ts are zero. Hence, we can use duality to write the optimal contracting problem as maximizing the utility of the manager subject to a nonnegativity constraint on pro…ts. Note that we write the nonnegativity constraint on pro…ts as (16) using the assumption that the expected value of " is zero along with the other features of the optimal contract derived above. We summarize this discussion in a proposition. Proposition 3. If AL < 1 the optimal contracting problem in a competitive equilibrium can be written as (14). Next, we will say that allocations are ex post ine¢ cient if " > ". If this inequality holds, then clearly all agents economy can be made better o¤ ex post by continuing the project in the states ["; " ]. Nonetheless, committing to ex post ine¢ cient allocations may be desirable as a way of providing the manager with stronger incentives for providing high e¤ort and thereby raising ex ante welfare. We now give su¢ cient conditions so that the optimal allocations with commitment require ex post ine¢ ciency. In providing these conditions, it is convenient to adopt a change of variables so that the manager can be thought of as choosing the probability of success p and incurring an e¤ort cost a(p). Formally, let a(p) be the inverse of the function pH so that a(p) = pH1 (p): Consider the allocations that arise when " is restricted to equal "; so that there is no ex post ine¢ ciency (no bankruptcy). Let pE H denote the optimal probabilities under this restriction. Proposition 4. If R is su¢ ciently close to 1 and a00 (pE H ) is su¢ ciently small then " > ": 13
That is, supporting ex ante e¢ cient allocations requires ex post ine¢ ciency. The proof of this proposition is in the appendix. The basic idea is that the incentive e¤ects associated with bankruptcy are large when a00 (p) is small. To see the role of these incentive e¤ects consider the …rst order condition associated with the incentive constraint, given by
(17) cH + BH(" ) = a0 (pH )
Consider the incentive gains from a small increase in the probability of bankruptcy resulting from an increase in " , holding …xed cH : Di¤erentiating (17) gives
Bh(" ) dpH = 00 d" a (p)
Thus, when a00 (p) is small the incentive gains from increasing the probability of bankruptcy are large. If R is su¢ ciently close to 1, the resource costs of increasing the probability of bankruptcy are small. Hence, when these conditions are met, supporting e¢ cient allocations requires a positive probability of bankruptcy. B. Implementing the competitive equilibrium Here we argue that the equilibrium outcomes can be interpreted as outcomes that arise with …nancial contracts that resemble debt, equity, and managerial compensation combined with an institutional arrangement that resembles bankruptcy. Under our interpretation the model implies a unique level of debt and equity. In this sense, the agency problems in our model make the Modigliani-Miller theorem not apply. 14
Our model implies a unique level of compensation for managers and a unique level of state-contingent payments to investors. Consider the following interpretation of these state contingent payments. Under this interpretation a …rm operated by a manager issues the following …nancial claims. The …rm issues (risky) debt and equity and enters into a compensation contract with the manager. The debt promises a face value of AL (1 + " ): The nature of the debt contract is that if the …rm is unable to meet the face value of its debt payments, the …rm is forced into bankruptcy, equity holders lose their claims and debt holders receive the liquidation value of the …rm, so that for each "
" debt-holders receive
RAL (1 + ")g(kc ). The manager’s compensation contract speci…es a payment of cH if the manager retains his managerial capability and if the …rm is successful and zero otherwise. Outside equity is the residual claimant. Suppose that the equilibrium allocation satis…es
(18) AH (1 + ")
cH
AL (1 + " )
and
(19) R
X
pi (a)Ai
AL (1 + " )
Note that (18) guarantees that in the high state when the manager keeps the ability to manage the project, the …rm can pay the face value of the debt, while (19) guarantees under the event that the manager loses the ability to manage the project, the …rm can pay the face value of the debt by selling its assets.
15
C. Without commitment Suppose now that the agents in this economy cannot commit to contracts. We show that equilibrium allocations without commitment give lower welfare than those with commitment. Speci…cally, suppose that after the action a has been taken and the …rst stage investments have been made, but before the state and the realization of " have occurred, …nancial intermediaries and managers can renegotiate their contracts if both parties agree. Clearly, all projects will be continued in order to avoid the output and the nonpecuniary costs of bankruptcy. To see this result more formally, suppose now that a manager has taken an action a and …rst stage investment decisions have been made, but uncertainty has not yet been realized. Consider the outcomes if the …rm and the manager agree to renegotiate. We model the renegotiation as follows. The …nancial intermediary makes a take it or leave it o¤er to the manager. If the manager takes the o¤er that o¤er is implemented, while if the manager rejects the o¤er the existing contract is implemented. Clearly, the manager will accept any o¤er which makes the manager at least as under the existing contract. Since the action a has already been taken, it is optimal for the …nancial intermediary to set " = " and avoid bankruptcy. In sum, in this static model without commitment the incentive to renegotiate is so strong that the equilibrium has no bankruptcy and, hence, no ex post ine¢ ciency. Thus, without commitment the optimal contracting problem solves (14) subject to the additional constraint that " = ". Clearly, welfare in an equilibrium without commitment is lower than that with commitment. 16
2. The Dynamic Contracting Model Here we develop a dynamic contracting model without commitment. We show that this lack of commitment constrains the optimal contracts entered into by private agents, relative to an environment with commitment. Our dynamic model is an in…nite repetition of a modi…ed version of our simple model. The main point of these modi…cations is to allow for …re sale e¤ects in which changes in the aggregate incidence of bankruptcy alter the prices at which assets are sold. In later sections when we turn to optimal policy these …re sale e¤ects will play a prominent role. A. The benchmark economy The benchmark economy we consider is an in…nitely-repeated version of a static model. Our benchmark economy has no technology to transform goods from period t to period t + 1; so that agents cannot save across periods. The static model is a version of the simple economy with three modi…cations. These three modi…cations allows for …re sale e¤ects. First, we assume that managers stochastically lose their ability to convert capital goods into consumption goods. Speci…cally, with probability
0
the capital goods produced in stage
2 can no longer be managed by the incumbent manager and must instead be used in the traditional technology. Second, we allow for an intensive margin in the corporate technology. Speci…cally, rather than restricting the scale kc of the corporate investment to be 1 we allow it vary. In particular, the amount of capital goods produced in stage 2 is
Ai (1 + ")g(kc ) for i 2 fH; Lg
17
where g is an increasing concave function with g 0 (0) …nite. Third, we replace the traditional technology which previously was simply described by the constant R < 1 with a constant returns to scale production technology F (k1 ; k2 ) where k1 denotes that capital invested in this technology in stage 1 by the lenders and k2 denotes the capital invested in this technology in stage 2. We assume that F is concave and has diminishing marginal products. We also assume that the incumbent managers are more productive in converting capital goods to consumption goods than is the traditional technology. That is, we assume that marginal product of k2 in the traditional technology is always less than the marginal product of capital in the corporate sector. Formally, F2 (k1 ; 0) = 1 so that F2 (k1 ; k2 )
1 for all k1 ; k2 :
The capital k2 invested in the traditional technology comes from two sources: the exogenously liquidated capital and the capital from bankrupt …nancial intermediaries and is given by
(20) k2 =
0
X
pi
Z
Ai (1 + ")dH(") g(kc ) +
1
X
pi
Z
Ai (1 + ")dH(") g(kc )
Bi
Here competitive …rms operate the traditional technology and choose k1 and k2 to maximize
F (k1 ; k2 )
R1 k1
R2 k2
The …rst order conditions are
(21) F1 (k1 ; k2 ) = R1
(22) F2 (k1 ; k2 ) = R2 18
The lenders in this economy choose how much of their endowment e to invest in the corporate technology, kc at rate Rc , how much to invest in the traditional technology, k1 at rate R1 , and how much to store, ks at rate 1. That is, lenders solve
(23) cI = max Rc kc + R1 k1 + ks
subject to
(24) kc + k1 + ks
e:
We will assume that all three technologies are used in equilibrium. A set of su¢ cient conditions is the following. First, e is su¢ ciently large, so that the storage technology is always used. Second, that the corporate technology is su¢ ciently productive in that AH is large enough and that p0H (0) is su¢ ciently large, so that it is always used. Finally, that F1 (0; k2 ) > 1 for all k2 > 0; so that the traditional technology is always used. Under these assumptions we have that
(25) Rc = R1 = 1
and we will impose this condition from now on. The resource constraint for this economy is
(26)
1 pH (a)cH
+c
I
1
pH (a)AH + pL (a)AL
Z
"
19
"
(1 + ")dH(") g(kc ) + F (k1 ; k2 )
With Commitment To set the stage for our environment without commitment by private agents, we brie‡y describe the dynamic model with commitment by private agents. In our model, …nancial intermediaries live for only one period and …nancial intermediaries in any period t cannot observe the output of …nancial intermediaries in earlier periods. Hence, managers cannot enter into contracts that condition on their past output levels. This assumption ensures that the manager’s incentive problem is static and that equilibrium in the dynamic model reduces to an in…nitely-repetition of that in the static model. Recall that in the simple economy, the incentive constraint for the manager is binding if AL < 1: It is straightforward to check that the incentive constraint in the benchmark economy is binding if AL ,
0
and g(e) are su¢ ciently small. We will assume that the incentive
constraint is binding in the benchmark economy from now on. We now set up the contracting problem for this economy. Following the logic of Proposition 4, the contracting problem solves
(27) max
1
[pH (a)cH
pL (a)BH(" )]
a
subject to
(28) a 2 arg max a
(29)
1
[pH (a)cH
1 pH (a)cH + kc
1
pL (a)BH(" )]
pH (a)AH + pL (a)AL
a: Z
"
20
"
(1 + ")dH(") g(kc ) + R2 k2
where
(30) k2 =
0
hX
i
pi (a)Ai g(kc ) +
1 pL (a)AL g(kc )
Z
"
(1 + ")dH("):
"
Recalling that in any equilibrium (25) holds, we have the following de…nition. A competitive equilibrium with commitment is an allocation cH ; a; " ; k1 ; k2 ; R2 , such that i) given R2 ; the allocations solve the contracting problem (27). ii) given R2 ; k1 and k2 satisfy (21) and (22). iii) the consumption of lenders satis…es (23). iv) the resource constraints (26) and (24) hold.
Without Commitment by Private Agents Without commitment by private agents, we require that the contracts managers and …nancial intermediaries enter into must be self enforcing. We say that a contract is selfenforcing if, after the manager has chosen the e¤ort level, the payo¤ from continuing with the contract is at least as large as the payo¤ from deviating. In order to construct the payo¤ associated with a deviation, we assume that if a deviation has occurred in any period, the payo¤s to the manager in all subsequent periods is given by the solution to the optimal contracting problem (27) with " = 0. Let U N denote the value of the contracting problem with this restriction. Under this assumption, it should be clear that if a manager and the …rm choose to deviate in some period t, they should choose a deviation that maximizes current payo¤s. As
21
in the simple economy without commitment, the best one-shot deviation is clearly to set " to zero to avoid the output and nonpecuniary costs of bankruptcy. Under the best one shot deviation the current period expected payo¤s to the manager are
(31) U^ (a; kc ) =
cH 1 pH (a)^
a=
1
[pH (a)AH + pL (a)AL ] g(kc ) + R2 k^2
kc
a
where c^H is the consumption associated with the renegotiated contract and
k^2 =
0
X
pi (a)Ai g(kc ):
For some given contract a; kc ; " if there is not deviation, then the manager’s expected consumption is determined from (29) and the manager’s payo¤s are given by
(32) U (a; " ; kc ) =
=
1
1 pH (a)cH
pH (a)AH + pL (a)AL
1 BH("
Z
)
a
"
(1 + ")dH(") g(kc ) + R2 k2
"
where
k2 =
0
hX
i
pi (a)Ai g(kc ) +
1 pL (a)g(kc )
Z
"
22
"
(1 + ")dH("):
1 BH("
)
kc
a
Given a continuation value U; we say that a contract is privately sustainable if
(33) U (a; " ; kc ) +
1
U
U^ (a; kc ) +
1
UN:
The optimal contracting problem without commitment is now to maximize the manager’s utility (27) subject to (28), (29), (30) and (33). A privately sustainable equilibrium is an allocation cH ; a; " ; k1 ; k2 , a price R2 ; and a continuation utility U such that i) given R2 ; the allocations solve the optimal contracting problem without commitment. ii) given R1 and R2 ; k1 and k2 satisfy (21) and (22). iii) given Rc ; R1 and kc = 1 , the consumption of lenders satis…es (23). iv) the continuation utility U equals U (a; " ; kc ): v) the resource constraints (24) and (26) hold. One rationalization for our formalization of the optimal contracting problem without commitment is that manager and …rm behavior is disciplined by trigger strategies. Under this rationalization, the optimal contracting problem …nds the best trigger strategy equilibrium in a game between the manager and …nancial intermediaries, holding …xed the prices in a competitive equilibrium. A standard result in the game theory literature is that the best equilibrium can be supported by a trigger strategy which prescribes the worst equilibrium continuation payo¤ following any deviation. In our economy, the worst equilibrium is the in…nite repetition of the static equilibrium without commitment. This in…nite repetition has per period value U N . 23
Under this rationalization, we assume that managers are in…nitely-lived but all agents in future periods only observe whether or not the manager has renegotiated in the past. This assumption keeps the manager’s incentive constraint static and allows us to focus on the incentives to renegotiate. Consider the following trigger strategies: if a manager ever renegotiates, then all …nancial intermediaries believe that the manager will always renegotiate so that bankruptcy will never be declared in the future. Since this continuation yields the worst payo¤s, it follows that the best equilibrium for the game between managers and …nancial intermediaries, holding …xed market prices, solves the optimal contracting problem. We emphasize that our notion of equilibrium does not depend on this rationalization. Formally, our optimal contracting problem is analogous to that in the literature on models with enforcement constraints, in that we replace the enforcement constraints by sustainability constraints. We now turn to welfare with and without commitment. We begin by showing that the equilibrium value of R2 is the same in the economies with and without commitment. To show this result note that in both economies F1 (k1 ; k2 ) = 1 and hence since F has constant returns to scale, this implies that F1 (k1 =k2 ; 1) = 1 so that k1 =k2 is the same value, say k~ ~ 1) we know R2 is also the same in both in both economies. Since R2 = F2 (k1 ; k2 ) = F2 (k; economies. We record this result in the following lemma. Lemma 1. The equilibrium values of R1 and R2 are the same in the economies with and without commitment. Furthermore, the value of R1 = 1. Since market prices are the same in the economies with and without commitment, the only di¤erence between the associated contracting problems is the private sustainability constraint. If this constraint is binding in the contracting problem, the privately sustainable 24
equilibrium yields lower welfare than the competitive equilibrium under commitment. The private sustainability constraint is binding if the discount factor by
is not too large. We denote
the critical value of the discount factor such that the the private sustainability constraint
just binds at the commitment allocations. That is
(34a) U (ac ; " c ; kcc ) +
where ac ; "
c
1
satis…es
U (ac ; " c ; kcc ) = U^ (ac ; kcc ) +
1
UN
denote the contract in a competitive equilibrium with commitment. Clearly, if
; the commitment outcomes are privately sustainable, and if
< ; the commitment
outcomes are not sustainable.
3. Adding Government Policies We now allow for the possibility of intervention by benevolent government authorities without commitment. We begin with a bailout authority which uses lump sum taxes and transfers to alter bankruptcy decisions. After managers have chosen their actions, the bailout authority has an incentive to use taxes and transfers to reduce ex post ine¢ ciency. In using these instruments, we assume that the bailout authority faces a trade o¤ parallel to that faced by private agents: if the authority deviates from some given equilibrium policy, private agents believe that the bailout authority will choose future policies so as to eliminate ex post ine¢ ciency. These beliefs induce a government sustainability constraint which is similar to the private sustainability constraint with one important di¤erence. This di¤erence is that the government sustainability constraint is tighter because it takes into account …re sales e¤ects.
25
That is, when a bailout authority intervenes to prevent bankruptcies ex post it recognizes it recognizes that its action raise the price of liquidated assets. In contrast, the actions of individual private agents do not a¤ect prices. In our model a rise in the price of liquidated assets raises welfare and therefore makes the government sustainability constraint tighter and hence makes the equilibrium outcomes with a bailout authority worse than without such an authority. We then ask, Can a regulator armed with the ability to limit the terms of private contracts improve on these outcomes? We …nd that it can. We show that the optimal regulation imposes a cap on the size of the corporate technology, a too-big-to-fail-cap, and a cap on the liquidation level, a bankruptcy cap. Such a regulator takes into account the incentives of the bailout authority to intervene and structures the terms of private contracts so as to reduce the incentives of the bailout authority to intervene. We show that the regulator can improve upon the equilibrium outcomes with a bailout authority. A. A Bailout Authority Consider a bailout authority that can make transfers or levy taxes on …nancial intermediaries contingent on the state and the realization of the idiosyncratic shock ": Suppose now that the bailout authority, as well as private agents, cannot commit to their future actions. The bailout authority’s per period payo¤ is given by the sum of the consumption of all agents in the economy. The bailout authority makes its policy decision after the managers have chosen their actions but before the realization of either the state, H or L or the shocks ": The instruments available to the bailout authority are a tax rate
in the high state and
lump sum transfers TL (") in the low state which are conditional on the …rm not declaring
26
bankruptcy in the low state when " is realized. The bailout authority’s budget constraint is
(35) pH = pL
Z
"
TL (")dH("):
"
After the bailout authority chooses its policy, private agents decide whether or not to accept the state contingent transfers TL (") in the low state and have no choice but to pay the taxes in the high state. We de…ne a sustainable equilibrium in the appendix. The basic idea is the same as in Chari and Kehoe (1990), in which the decisions of the private agents and the government are functions of the history of their past decisions. Here we will show that any sustainable equilibrium must satisfy a government sustainability constraint which is analogous to the private sustainability constraint. In order to develop this constraint, consider the current payo¤ of the government when all contracts are renegotiated. This payo¤ is given by
(36) U^ G (a; kc ) =
where k^2 =
0
P
1
[pH (a)AH + pL (a)AL ] g(kc ) + F (k1 ; k^2 ) + e
kc
k1
a
pi (a)Ai g(kc ):
A sustainable equilibrium is de…ned in the We now develop the bailout authority’s sustainability constraint. We will argue that As in the environment without commitment by private agents, we begin by characterizing the equilibrium in which after any deviation, agents believe that all future contracts will be renegotiated and hence revert to an equilibrium with " = 0: The reversion equilibrium has per period value U N as before. The only subtlety to keep in mind is that, from Lemma 27
1, R2 has the same value as in the static economy with commitment. Consider the best one shot deviation for the bailout authority. It is clearly optimal for the authority to set policy so that the economy has no bankruptcy. PUT IN THAT IT IS FEASIBLE In such a case, given some value of k1 ; the sum of utility of managers and lenders is given by ): If the bailout authority chooses not to deviate from some given contract then the sum of utility of managers and lenders is given by U G (a; " ; kc ) which equals
1
pH (a)AH + pL (a)AL
Z
"
(1 + ")dH(") g(kc )
1 pL (a)BH("
)+F (k1 ; k2 )+e kc k1 a
"
where
k2 =
0
hX
i
pi (a)Ai g(kc ) +
1 pL (a)g(kc )
Z
"
(1 + ")dH(")
"
Note that the continuation payo¤ if the government chooses not to deviate is the same as that in (32). Given a continuation utility U G ; we say that a contract is sustainable to bailouts if
(37) U G (a; " ; kc ) +
1
UG
U^ G (a; kc ) +
1
UN:
A policy induces a competitive equilibrium as follows. Given a policy, the budget
28
constraint of the …nancial intermediary becomes
(38) 1 pH (a)cH + kc
1
pH (a)(AH
) + pL (a)
Z
"
[AL (1 + ") + TL (")] dH(") g(kc ) + R2 k2 :
"
The optimal contracting problem with a bailout policy is to choose a contract cH ; a; kc and " to maximize the utility of the manager (27) subject to the incentive constraint for the manager (28), the private sustainability constraint (33), and the budget constraint of the …nancial intermediary (38) where k2 is given by (30) A sustainable equilibrium with a bailout policy consists of an allocation cH ; a; " ; k1 ; k2 ; kc ; R2 , U; U G and a policy ; TL (") such that i) given R2 ; the allocations solve the optimal contracting problem with policy ii) given R2 ; k1 and k2 satisfy (21) and (22) iii) the consumption of lenders satis…es (23) with Rc = R1 = 1: iv) the resource constraints (26) and (24) hold. v) the government’s budget constraint (35) holds. vi) the government’s sustainability constraint (37). vii) the continuation utility U = U (a; " ; kc ) and U G = U G (a; " ; kc ) We then have the following proposition. Proposition 5. Consider any contract (a; " ; kc ) with " > " and suppose that F (k1 ; k2 ) is strictly concave in k2 . The government sustainability constraint (37) is tighter than the private sustainability constraint (33), in the sense that if any such contract satis…es (37) it also satis…es (33). Furthermore, if any such contract satis…es (33) with equality, it violates 29
(37). Proof. From Lemma 1 it follows that the continuation utility following a deviation U N in the private sustainability constraint is the same as it is in the government sustainability constraint. Thus, we need only show that
(39) U^ G (a; kc )
U G (a; " ; kc ) > U^ (a; kc )
U (a; " ; kc )
From Euler’s theorem F (k1 ; k2 ) = F1 k1 + F2 k2 . Since F1 = 1 in any equilibrium and F2 = R2 it follows that
(40) F (k1 ; k2 )
k1 = R2 k 2
Using (40) it follows that U G (a; " ; kc ) = U (a; " ; kc )+e: Using this result and canceling terms in (39) gives that (39) holds if and only if
(41) F (k1 ; k^2 )
k1 > R2 k^2
Adding R2 k2 to both sides, using Euler’s theorem and rearranging terms, (41) can be written as
(42) R2 (k2
k^2 ) > F (k1 ; k2 )
F (k1 ; k^2 )
Since " > " it follows that k2 > k^2 . Hence, (42) must hold because F is a strictly concave function of k2 ; . This result proves that (37) is tighter than (33). Q:E:D: 30
If the production function satis…es (42) we say that the economy has …re sale e¤ects. The key idea in the proof of Proposition 8 is that when the bailout authority contemplates a deviation it realizes that by lowering the measure of bankruptcies, it recognizes the e¤ects of …re sales. That is, it recognizes that lowering the measure of bankruptcies raises the value R2 of the capital that is transferred from the corporate sector to the traditional sector. In contrast, when a private …rm contemplates a deviation it takes the value R2 as given. Thus, the right side of the private sustainability constraint is lower than the right side of the sustainability to bailout constraint. Note that if there are no …re sale e¤ects the private sustainability constraint and the government sustainability constraint coincide. To see this suppose that F is linear in k1 and k2 so that it can be written as F (k1 ; k2 ) =
1 k1
+
2 k2
where
1
and
2
are constants. Then
it is easy to show that
U^ G (a; kc )
U G (a; " ; kc ) = U^ (a; kc )
U (a; " ; kc )
so that the two constraints coincide. We use Proposition 5 to show that the sustainable equilibrium with bailouts yields lower welfare than the privately sustainable equilibrium. Proposition 6. Suppose the discount factor
is strictly less than the threshold
given
by (34a) at which the private sustainability constraint is binding. Any sustainable equilibrium with bailouts yields strictly lower welfare than the privately sustainable equilibrium. Furthermore, any sustainable equilibrium with bailout policy has bailouts in equilibrium, in the sense that
> 0. 31
Proof. Since
0 by way of
= 0. Then, using Lemma 1 it follows that the solution to the
dynamic contracting problem coincides with that of the privately sustainable equilibrium. This allocation violates the government sustainability constraint. Thus, any sustainable equilibrium with bailout policy must have
> 0: Q:E:D:
Characterization of the best bailout equilibrium: Let
V ("b ) = max U (a; " ; kc )
subject to (28), (29), (30) and the additional constraint
" = "b :
32
Now consider the maximization problem
max V ("b )
subject to (37). The solution to this problem consists of the best baiout equilibrium allocations. (INSERT PROOF)
B. Can an ex ante regulator improve welfare? Consider the situation described in the previous section in which neither the bailout authority nor the private agents can commit to their actions. We show that a regulatory authority armed with the ability the dictate the terms of the private contract, namely the R compensation contract cR H , the scale of the corporate technology kc ; and the liquidation level
"R , can improve ex ante welfare. Such a regulator must take into account the incentives of the bailout authority to intervene. To see how a regulator can improve upon equilibrium allocations, we need to de…ne a competitive equilibrium with regulation. We begin with an extreme form of regulation in which the regulator speci…es the exact size of the …rm and the exact set of states in which the …rm can declare bankruptcy, and then show that less extreme regulations can achieve desired outcomes. Under the extreme form of regulation, the regulator chooses taxes, transfers and speci…es the following constraints on contracts.
(43) kc = k r and " = "r :
The optimal contracting problem with regulation is now to choose a contract cH and " to 33
maximize the utility of the manager (27) subject to the incentive constraint for the manager (28), the private sustainability constraint (33), the budget constraint of the …nancial intermediary (38) where k2 is given by (30) and subject to the policy constraints (43). A sustainable equilibrium with regulation consists of an allocation cH ; a; " ; k1 ; k2 ; kc R2 ,U and a regulatory policy k r ; crH ; "r ; ; TL (") is de…ned is de…ned in the same way as a sustainable equilibrium with bailout policy with one important di¤erence. That di¤erence, of course, is that the contracting problem now has additional constraints. The regulator’s problem is to structure policies so as to maximize the manager’s welfare given that the allocations associated with a given policy must be part of a sustainable equilibrium. Consider the regulator’s problem given utility level e for lenders is to choose cH ; a; " ; kc ; k1 ; k2 ; ks to solve
(44) max
1
[pH (a)cH
pL (a)BH(" )]
a
subject to the manager’s incentive constraint
(45) a 2 arg max a
1
[pH (a)cH
pL (a)BH(" )]
a
the resource constraint
(46)
1 pH (a)cH
+c
I
1
pH (a)AH + pL (a)AL
Z
"
34
"
(1 + ")dH(") g(kc ) + F (k1 ; k2 ) + ks
where k2 is given by
(47) k2 =
0
hX
i
pi (a)Ai g(kc ) +
1 pL (a)g(kc )
Z
"
(1 + ")dH(")
"
voluntary savings by lenders
(48) F1 (k1 ; k2 ) = 1
and the bailout authority’s sustainability constraint
(49)
U (a; " ; kc ) 1
U^ G (a; kc ) +
1
UN
minimum utility level for managers
(50) cI
e
the stage 1 investment constraint
(51) kc + ks + k1
e:
Note that the voluntary savings by lenders constraint (48) arises because the regulator has no instruments that can a¤ect the return to investment k1 in the traditional technology. The regulatory problem is equivalent to the simpli…ed regulatory problem of maximizing
35
(44) subject to (45),
(52)
1 pH (a)cH + kc
1
pH (a)AH + pL (a)AL
Z
"
(1 + ")dH(") g(kc ) + R2 k2
"
where k2 is given by (47), and (49). To see these two problems are equivalent, we show that the constraint sets of the two problems are the same. Consider …rst showing that (52) is implied by the constraints of the original problem. To do so, substitute cI = e = kc + ks + k1 into (46), use F1 = 1 and Euler’s theorem to obtain (52). Since the rest of the constraints are the same, it follows that if an allocation satisi…es the constraints of the regulatory problem then it satis…es the constraints of the restated problem. Now consider an allocation that satis…es the constraints of the restated problem. We need to show that k1 and ks can be chosen to satisfy the original problem. To do so de…ne
k1 = k2
where
is such that F1 (k1 ; k2 ) = F1 ( ; 1) = 1 and choose ks = e
kc
k1 . Since we have
assumed that e is large, a nonnegative ks can be so chosen. Thus, the constraint sets are equivalent. Proposition 7. The solution to the simpli…ed regulator’s problem coincides with the best sustainable equilibrium with regulation and has taxes equal to zero. Proof. Inspection of the problems and constraints that de…ne the sustainable equilibrium make it clear that any such equilibrium must satisfy (45)-(51). Clearly, the regulatory
36
equilibrium must maximize manager’s welfare subject to these constraints. Next we show that any solution to the simpli…ed regulator’s problem can be supported as a sustainble equilibrium with regulation. To do so in the sustainable equilibrium let the policies be de…ned by (43) and let
= TL (") = 0: Clearly, the simpli…ed regulator’s problem
is equivalent to one in which we replace the government sustainability constraint (49) with the (43). From Proposition 5 the private sustainability constraint is necessarily satisi…ed at these allocations. Thus, the equivalent problem coincides with the optimal contracting problem without commitment. Q:E:D: Next, we have Proposition 8. Suppose
< : The regulatory equilibrium yields higher welfare than
any sustainable equilibrium with bailout policy. Proof. The proof is by contradiction. Suppose that the bailout authority could achieve the same allocations as the regulator. Since the sustainability constraint for the government is tighter than it is for private agents, then at the regulator’s allocations the private sustainability constraint in the contracting problem must be slack. Now consider the …rst order conditions with respect to kc in the regulator’s problem and in the contracting problem. To derive these …rst order conditions for the regulator’s problem we …rst rewrite the resource constraint (46). To do so we note from Euler’s theorem that F = F1 k1 + F2 k2 ; so that using (48) we have that F = k1 + R2 k2 , where, as before, R2 is uniquely pinned down by the condition that F1 = 1: Substituting F = k1 + R2 k2 and using that (50) and (51) hold with equality we can rewrite this constraint as
1 pH (a)cH
1
pH (a)AH + pL (a)AL
Z
"
(1 + ")dH(") g(kc ) + R2 k2
"
37
kc
In the rewritten regulator’s problem the …rst order condition for kc is given by
0
(53)
fg (kc ) [Yc + R2 Yl ]
where
and
Yc =
1
h
1g =
U^kG (a; kc )
i Uk (a; " ; kc )
are the multipliers on (46) and (49),
pH (a)AH + pL (a)AL
Z
"
(1 + ")dH(")
"
Yl =
0
[pH (a)AH + pL (a)AL ] +
1 pL (a)AL
Z
"
(1 + ")dH(")
"
Consider next the …rst order conditions for the dynamic contracting problem in dual form with a slack private sustainability constraint. The budget constraint for this problem is given by
1 pH (a)cH
1
pH (a)(AH
) + pL (a)
Z
"
[AL (1 + ") + TL (")] dH(") g(kc ) + R2 k2
"
The …rst order condition for kc for this problem is given by
(54) ~ fg 0 (kc ) [Yc ( ; TL ) + R2 Yl ]
1g = 0
38
kc
where
Yc ( ; TL ) =
1
pH (AH
) + pL
Z
"
[AL (1 + ") + TL (")] dH(") :
"
In equilibrium, the government’s budget constraint implies the value of taxes equal the value of transfers so that
(55) Yc = Yc ( ; TL ):
Combining (54) and (55) and using that the multiplier on the budget constraint is nonzero gives that in bailout equilibrium the allocations satisfy
(56) fg 0 (kc ) [Yc + R2 Yl ]
1g = 0
The bailout allocations do not satisfy the …rst order condition for the regulatory equilibrium (53) because the right side of (53) is nonzero. Hence, the allocations in the regulator’s problem and the contracting problem must di¤er. Since the bailout allocations are feasible for the regulator’s problem, the bailout equilibrium must yield lower welfare than the regulatory equilibrium. Q:E:D: The idea behind this proposition is that since the bailout authority has a balanced budget, in equilibrium, the tax-transfer scheme can only indirectly in‡uence the choice of kc . The key idea is that the regulator has a richer set of policy instruments than does the bailout authority.
39
C. A simple implementation We consider how we can implement the regulatory equilibrium as a competitive equilibrium in which the goverment imposes constraints on the type of contracts that …rms sign with managers. In the next proposition we show that the solution to the regulator’s problem can be implemented a too-big-to-fail cap of the form kc
k r and a liquidation constraint of
"r ; with no taxes or transfers.
the form "
Proposition 9. The solution to the regulator’s problem can be implemented with a too-big-to-fail cap of the form
(57) kc
kr
and a liquidation cap of the form
(58) "
"r
where "r is the optimal level of " in the solution to regulator’s problem. The proof is in the appendix. Next we show that the regulatory equilibrium can be implemented by a cap on the size of the …rm kc and a cap on the …rm’s debt to value ratio. From Proposition 9, it follows that to show this result we need only show that a cap on the …rm’s debt to value ratio is equivalent to a cap on " : We start by calculating the …rm’s debt to value ratio under this decentralization. To do so we calculate the expected present value of debt payments. With probability
0,
the manager loses the ability to manage the …rm and the debt holders receive 40
the face value of their debt. With probability
1 (pH (a)
+ pL (a)(1
H(" )) +
0;
the …rm’s
cash ‡ows exceed the required debt payment. In the event of bankruptcy, debt holders receive the liquidation value of the debt. The present value of debt payments is then given by
(59) f
1
[pH (a) + pL (a)(1
H(" ))] +
0 gAL (1+"
)g(kc )+
1 pL (a)R2 AL g(kc )
Z
"
(1+")dH("):
"
The value of the …rm is simply kc : We now argue that if R2 is su¢ ciently close to 1 then debt to value ratio is increasing in " . To see this result we note that the derivative of (59) with respect to " is proportional to
f
1
[pH (a) + pL (a)(1
H(" ))] +
0g
1 (1
R2 )pL (a)h(" )(1 + " )
Clearly, if R2 is close enough to 1 then the debt is increasing in " . Letting D denote the value of debt under any contract and Dr denote the value of debt (59) in a regulatory equilibrium, we summarize this discussion in a proposition. Proposition 10. If R2 is su¢ ciently close to 1 then the solution to the regulator’s problem can be implemented with a too-big-to-fail cap of the form kc debt to value ratio of the form
D kc
Dr : kr
41
k r and a cap on the
4. Conclusion We have made three points in this paper. First, ex ante e¢ cient contracts often require ex post ine¢ ciency. Second, the time inconsistency problem for the government is more severe than for private agents because …re sale e¤ects give governments stronger incentives to renegotiate contracts than private agents. Third, given that the government cannot commit itself to not bailing out …rms ex post, ex ante regulation of …rms is desirable.
42
5. Appendix Proposition 4. If R is su¢ ciently close to 1 and a00 (pE H ) is su¢ ciently small then " > ": That is, supporting ex ante e¢ cient allocations requires ex post ine¢ ciency. Proof: The proof is by contradiction. Suppose that " = ": We will show that a small increase in " from " raises welfare. To show this result, we totally di¤erentiate the budget constraint (16) and the incentive constraint (17) and evaluate these derivatives at " = ": We obtain the following relationships between dcH ; dp and d"
(60) [AH
AL
cH ] dp
pdcH
(1
p)AL (1
R2 )h(")d" = 0
(61) dcH + Bh(")d" = a00 (p)dp
where p = pH : Substituting for dp from (61) into (60) and rearranging terms we obtain
(62)
p
[AH
AL 00 a (p)
cH ]
dcH = [AH
AL
cH ]
B a00 (p)
The budget constraint evaluated at " = " implies that 1
(1
p)AL (1
AL = p(AH
R2 ) h(")d" :
AL
cH ); so that
(62) can be rewritten as
(63)
p
1 AL pa00 (p)
dcH =
1 AL B pa00 (p)
(1
p)AL (1
R2 ) h(")d"
Totally di¤erentiating the objective function, we obtain that the change in the utility of the
43
manager dU is given by
dU = pdcH
(1
p)Bh(")d"
and from (63) we have that dU > 0 if and only if
(64) dU = p
h
1 AL B pa00 (p)
(1
p)AL (1 1 AL pa00 (p)
p
i R2 )
(1
p)B > 0:
Next, we show that the denominator of the …rst term in (64) is positive. To do so, consider the solutions to cH and p obtained from the incentive constraint (15) and the budget constraint (16). Typically, these conditions yield multiple solutions. The solution that maximizes the manager’s welfare is the largest value of p that satis…es these conditions. Substituting for cH from (15) into (16) we obtain
(65) pa0 (p) + 1 = pAH + (1
p)AL :
At the largest value of p that satis…es (65), we must have that the derivative of the left side of (65) must be greater than the derivative of the right side of (65) so that
(66) pa00 + a0 (p) > AH
AL
Since the incentive constraint requires that a0 (pH ) = cH and the budget constraint implies
44
that 1
AL = p [AH
pa00 >
1
AL
cH ] ; (66) can be written as
AL : p
Thus, the denominator of the …rst term in (64) is positive. Next we rewrite (64) as 2
dU = 4p
h
1 AL pa00 (p)
i
1 AL pa00 (p)
p
3
B
p)5 B
(1
p(1
p)
AL (1
R2 ) 1 AL pa00 (p)
p
>0
which, in turn can be rewritten as 2h
(67) dU = 4
1 AL pa00 (p)
p
i
p(1 1 AL pa00 (p)
Since p < 1; p(1
p)
p)
3
5B
p(1
p)
AL (1 p
1=4 so that (1
R2 ) 1 AL pa00 (p)
AL )=pa00 (p)
>0
p(1
p) > 0 if a00 (p) < 4(1
AL ):
Thus if a00 (p) is su¢ ciently small, the …rst term in (67) is positive and if R2 is su¢ ciently close to 1 the second term is small, so that, under these conditions, the change in utility given in (67) is positive. Q:E:D: Proposition 9. The solution to the simpli…ed regulator’s problem can be implemented with a too-big-to-fail cap of the form kc
k r and a liquidation cap of the form "
"r where
k r and "r are part of the solution to the regulator’s problem. Proof. We will show that the simpli…ed regulatory problem is equivalent to a constrained regulatory problem, is which the sustainability constraint in the simpli…ed regulatory
45
problem are replaced by two constraints of the form
(68) kc
k r and "
"r :
To do so we consider an intermediate problem, call the constrained commitment problem which is competitive contracting problem (27) with the extra constraint
(69) kc
kr :
Note that the constraint (69) binds.(prove it binds) Let ("c ; ac ; k r ) denote the solution to this problem. We claim that
(70) "r
"c :
If the sustainability constraint (49) does not bind then clearly "r = "c . Suppose next that the sustainability constraint does bind. Then
(1
)U^ (ac ; k r ) + U N
(72) U (ar ; "r ; k r ) = (1
)U^ (ar ; k r ) + U N
(71) U (ac ; "c ; k r )
Next, since the regulatory solution is feasible for the constrained commitment problem
(73) U (ar ; "r ; k r )
U (ac ; "c ; k r ): 46
Thus, using (71)- (73) gives
(74) U^ (ar ; k r )
U^ (ac ; k r ):
From (31) we have that
(75) U^ (a; kc ) = (
1
+
0 R2 ) [pH (a)AH
+ pL (a)AL ] g(kc )
kc
a
Note that the U^ (a; kc ) is a strictly concave function of e¤ort which is increasing below its maximum and decreasing above its maximum. This function reaches its maximum at the full information optimum level of e¤ort aF I . We claim that both ar and ac are below aF I . To see this claim, suppose for example that ar is above aF I ; then it is feasible to reduce the costs for sustaining this e¤ort by reducing "r and achieve a higher level of utility. A similar argument holds for ac : Since ar and ac are both below aF I then U^ (a; kc ) is increasing in a in this region and it follows from (74) that
(76) ar
ac :
We next show that (76) implies that "r < "c . To do consider a version of the constrained commitment problem in which we replace (68) with
kc
k r and "
"
47
for an arbitrary " 2 ["; "] : Clearly, in any solution to this problem kc = k r . Denote the solution to this problem by a("). Since pH (a) satis…es the monotone likelihood ratio property as well as convexity of distribution function assumption the …rst order approach is valid. It follows immediately that the maximizer a(") is a continuous function of ". We next claim that a("c )
a(") for all " 2 ["; "c ] : To show this, suppose by way of
contradiction that for some " 2 ["; "c ] ; a(") > a("c ). This choice of " leads to higher output and higher utility than does ("c ; a("c )), which contradicts that the choice of "c maximizes utility. Next, note that a("r )
a("). To show this, suppose by way of contradiction that
a("r ) < a("). Then the choice ("; a(")) by the regulator yields higher utlity than ("r ; a("r )) and satis…es the sustainability constraint. This is a contradiction that ("r ; a("r )) solves the regulator’s problem. Since a(") is continuous and a("r ) 2 [a("); a("c )] then there exists some value of " 2 ["; "c ] such that a("r ) = a("). We claim that "r is the smallest value of " with this property. To see this claim note that any higher value of " with this property has lower output and lower utility than the smallest value does. We have therefore shown that "r
"c .
Note, for later use, that this argument also implies that utility is lower than the regulatory value for any value of " less than "r : We now establish our main result. To that end consider a contracting problem with a too-big-too-fail cap kc
k r and a liquidation cap "
"r . Clearly, the constraint kc
kr
binds. Since, as we have just remarked, utility is lower than the regulatory value for any value of " less than "r is follows that the solution to the contracting problem has " = "r . Q:E:D: 48
Appendix B: Setup and de…nition of a privately sustainable equilibrium The timing of events within a period are that …rst …nancial intermediaries simultaneously o¤er contracts to managers. The managers then decide whether to accept or decline the contracts. A manager who has accepted a contract with a …nancial intermediary then chooses a privately observed e¤ort level. After the e¤ort level has been chosen the …nancial intermediary then makes a take it or leave it o¤er to the manager. If the manager rejects renegotiated contract, the original contract is implemented. If the manager accepts then renegotiated contract then the renegotiated contract is implemented. The idiosyncratic state is then realized and payments occur according to the implemented contract. (Note that we have abstracted from renegotiation with lenders. Adding in such renegotiation simply adds notation with no substantive change in the results.) Lenders and the …nancial intermediary in period t observe the past history of a managers renegotiation choices, Ht = (Ht 1 ; t ), but do not observe the past realizations of output2 . We refer to Ht as the public history of a manager. Let
xt (Ht 1 )
denote the period
t contract xt = (cHt ; "t ; kct ) o¤ered by the …nancial intermediary to a manager at t3 . Let hct = (Ht 1 ; cHt ; "t ; kct ) be the history given the contract. Let
at (hct )
denote the
e¤ort decision of the manager. Let hat = (hct ; at ) denote the history inclusive of the private action. Let
t
= 1 signify that a new contract is o¤ered and
contract is o¤ered. Let
t (hct )
t
= 0 signify that no new
denote the strategy for o¤ering a new contract. Let
2
rt (hct )
It is this latter feature that ensures that the only intertemporal connection in the contract is whether the manager renegotiates. In particular, since the contract cannot depend on realizations of past output, this feature ensures that reputational outcomes in which the manager is induced to supply high e¤ort by a promise of better contracts in the future are not feasible. 3 Note the here we are focusing on equilibria in which the continuation payo¤s depend only on the public history.
49
denote the terms of the continuation contract (^ cHt ; ^"t ), if one was o¤ered: Let the new contract is accpeted and
t
t
= 1 denote
= 0 denote that either the new contract is rejected or
that no new contract is o¤ered, and let
^Ht ; ^"t ; t (hat ; c
t)
denote the associated acceptance
strategy. If
^Ht ; ^"t ; t (hat ; c
t)
= 1; then the manager’s payo¤s for the current period are given
by U (cHt ; "t ; at ) de…ned by
pH (at )cHt
BH("t )
at
and the …nancial intermediary’s payo¤s are
[pH (at )AH +pL (at )AL
Z
t (cHt ; "t ; kct ; at )
"
(1+")dH(")+pL (at )AL R2t
"t
If
^Ht ; ^"t ; t (hat ; c
t)
Z
de…ned by
"t
(1+")dH(")]g(kct ) pH (at )cHt kct
"
= 0; then the manager’s payo¤s for the current period are given by
U (^ cHt ; ^"t ; at ) and the …nancial intermediary’s payo¤s are
cHt ; ^"t ; kct ; at ). t (^
A strategy pro…le induces allocations and continuation utilities in the usual fashion. Let Vt (Ht 1 (i)) denote the continuation utility of a manager with public history Ht 1 (i): We will say that a collection of strategies is a privately sustainable equilibrium if (i) For any history (hat ; c^Ht ; ^"t ;
t)
with
t
= 1; if the equilibrium strategy speci…es
(77) U (cHt ; "t ; at ) + Vt+1 (Ht 1 ; 0)
U (^ cHt ; ^"t ; at ) + Vt+1 (Ht 1 ; 1)
50
t
= 0 then
while if the equilibrium speci…es
t
= 1 then the inequality (77) is reversed,
(iia) for any history hct , i then the original contract (cHt ; "t ) must yield higher payo¤s than any alternative contract (^ cHt ; ^"t ) that would be accepted by the manager in that
t (cHt ; "t ; kct ;
at (hct ))
for all (^ cHt ; ^"t ) such that
t (hct ;
cHt ; ^"t ; kct ; t (^
at (hct ))
^Ht ; ^"t ; 1) at (hct ); c
= 1:
(iib) For any history hct , if the equilibrium strategy speci…es renegotiate in that t (hct )
= 1 and o¤er (^ cHt ; ^"t ) and the o¤er is accepted in that
t (hct ;
^Ht ; ^"t ; 1) at (hct ); c
=1
then this new contract must yield the highest payo¤ to the …nancial intermediary among all o¤ers that will be accepted, in that
cHt ; ^"t ; kct ; t (^
at (hct ))
for all (~ cHt ; ~"t ) such that
cHt ; ~"t ; kct ; t (~
~Ht ; ~"t ; 1) t (hat ; c
at (hct ))
= 1: Also, if must be optimal for the intermediary
to renegotiate in that
cHt ; ^"t ; kct ; t (^
at (hct ))
t (cHt ; "t ; kct ;
at (hct )):
(Clearly we can ignore strategies that specify renegotation and rejection since the …nancial intermediary could attain the same payo¤s by not renegotiating.) (iii) For any history hct ; if the equilibrium strategy speci…es do not renegotiate in that 51
t (hct )
= 0; then the manager’s strategy for e¤ort
at (hct )
2 arg max pH (a)cHt a
BH("t )
at (hct )
satis…es
a;
while if the equilibrium strategy speci…es renegotiate to a new contract (^ cHt ; ^"t ) then
at (hct )
satis…es
at (hct )
2 arg max pH (a)^ cHt a
BH(^"t )
a:
(iv) For any history Ht 1 ; an o¤ered contract xt together with the strategies
at ;
t;
rt ;
induces an implemented contracted x~t , an associated action a ~t (where the implemented contract is either the original contract or the renegotiated contract), and an associated accept decision ~t . Let
c~Ht (hct ); ~"t (hct ); kct ; a ~t (hct ); ~t (hct )
denote the induced outcomes. The equilibrium strategy
xt (Ht 1 )
must maximize the …nan-
cial intermediary’s pro…ts over all contracts that will be accepted by the manager in that
t
(~ cHt (hct ); ~"t (hct ); kct ; a ~t (hct ))
t
^ ct ); ~"t (h ^ ct ); k^ct ; a ^ ct ) c~Ht (h ~t (h
^ ct = (Ht 1 ; x^t ) for all alternative contracts x^t such that where h
^ ct ); ~"t (h ^ ct ); a ^ ct )) + Vt+1 (Ht 1 ; ~t (h ^ ct )) (78) U (~ cHt (h ~t (h
52
Vt (Ht 1 )
t
(v) For any history Ht 1 , the …nancial intermediary makes zero pro…ts in that
t
(~ cHt (hct ); ~"t (hct ); kct ; a ~t (hct )) = 0
where hct = (Ht 1 ;
xt (Ht 1 )):
(vi) For any history Ht 1 ; continuation utilities are generated by the equilibrium strategies in that
(79) Vt (Ht 1 ) = U (~ cHt (hct ); ~"t (hct ); a ~t (hct )) + Vt+1 (Ht 1 ; ~t (hct ))
and
lim sup
t!1
t
Vt (Ht 1 (i)) = 0:
wlog the induced contract is o¤ered. any equilibria Appendix C: Setup and de…nition of a sustainable equilibrium: add a gov’t Let xt (i) = (cH (i); a(i); " (i); kc (i); (i); c^H (i); ^" (i); Yt ) and let ht (i) = (ht 1 (i); xt (i)) where
t (i)
= 1 indicates that manager i renegotiated in period t and
t (i)
= 0 indicates that
manager i did not renegotiate in period t: Lenders and the …nancial intermediary in period t observe the past history of manager i0 s renegotiation choices, Ht (i) = (Ht 1 (i); t (i)), but do
53
not observe the past realizations of output4 . Let
it (Ht 1 (i))
denote the contract o¤ered by
the …nancial intermediary to a manager i at t: Let
M it (ht 1 (i); cHt (i); at (i); "t (i); kct (i))
denote the decision of the manager on whether to renegotiate. A strategy pro…le induces allocations and continuation utilities in the usual fashion. Let U (a(i); " (i); kc (i))) denote the period utility associated with a contract (a(i); " (i); kc (i))) and let V (Ht (i)) denote the present discounted continuation utility of a manager with public history Ht :5 We will say that a sequence of contracts and renegotiation decisions is a sustainable equilibrium if i) along the M it (ht 1 (i),
equilibrium path contract
(80)
X
it (Ht 1 (i))
pi (a)
Z
solves the …rm’s problem, maximize
Yci (")dH(") +
Ci
i
cHt (i); at (i); "t (i); kct (i))) = 0; ii) for all histories Ht 1 (i); the
Z
Ybi (")dH(")
Bi
Z
[ci (") + di (")]dH(")
subject to the participation constraint for lenders
(81)
X i
pi (a)
Z
di (")dH(")
1;
4
It is this latter feature that ensures that the only intertemporal connection in the contract is whether the manager renegotiates. In particular, since the contract cannot depend on realizations of past output, this feature ensures that reputational outcomes in which the manager is induced to supply high e¤ort by a promise of better contracts in the future are not feasible. 5 Note the here we are focusing on equilibria in which the continuation payo¤s depend only on the public history
54
the participation constraint for managers
(82)
X
pi (a)
i
Z
ci (")dH(")
B
Z
dH(")
a
U;
Bi
the managers’incentive constraint
(83) a 2 arg max a
X i
pi (a)
Z
ci (")dH(")
B
Z
dH(")
a
Bi
the nonnegativity constraint on consumption of managers
(84) ci (")
0
and the private sustainability constraint
(85) U (a(i); " (i); kc (i))) + V (Ht 1 (i); 0)
U^ (a(i); ^" (i); kc (i)) + V (Ht 1 (i); 1)
where we have suppressed the period t subscript on current allocations for convenience. The sustainability constraint in this contracting problem is needed to ensure that the manager does not have an incentive to renegotiate the contract. To see this result, suppose that a contract does not satisfy this constraint. Then the manager will renegotatiate the contract. The publicly observed history of manager i is Ht (i) = (Ht 1 (i); t (i)): A sustainable equilibrium with a bailout policy consists of an allocation cH ; a; " ; k1 ; k2 ; kc ; R2 , 55
U; U G and a policy ; TL (") such that i) given R2 ; the allocations solve the optimal contracting problem with policy ii) given R2 ; k1 and k2 satisfy (21) and (22) iii) the consumption of lenders satis…es (23) with Rc = R1 = 1: iv) the resource constraints (26) and (24) hold. v) the government’s budget constraint (35) holds. vi) the government’s sustainability constraint (37). vii) the continuation utility U = U (a; " ; kc ) and U G = U G (a; " ; kc )
56